Download Polymer Collapse, Protein Folding, and the Percolation Threshold

Polymer Collapse, Protein Folding, and the
Percolation Threshold
HAGAI MEIROVITCH
University of Pittsburgh, School of Medicine, Center for Computational Biology and Bioinformatics
(CCBB), Suite 601 Kaufmann Building, 3471 Fifth Avenue, Pittsburgh, Pennsylvania 15213
Received 21 March 2001; Accepted 22 May 2001
Abstract: We study the transition of polymers in the dilute regime from a swollen shape at high temperatures to their
low-temperature structures. The polymers are modeled by a single self-avoiding walk (SAW) on a lattice for which l
of the monomers (the H monomers) are self-attracting, i.e., if two nonbonded H monomers become nearest neighbors
on the lattice they gain energy of interaction ( = −||); the second type of monomers, denoted P, are neutral. This
HP model was suggested by Lau and Dill (Macromolecules 1989, 22, 3986–3997) to study protein folding, where H
and P are the hydrophobic and polar amino acid residues, respectively. The model is simulated on the square and simple
cubic (SC) lattices using the scanning method. We show that the ground state and the sharpness of the transition depend
on the lattice, the fraction g of the H monomers, as well as on their arrangement along the chain. In particular, if the H
monomers are distributed at random and g is larger than the site percolation threshold of the lattice, a collapsed transition
is very likely to occur. This conclusion, drawn for the lattice models, is also applicable to proteins where an effective
lattice with coordination number between that of the SC lattice and the body centered cubic lattice is defined. Thus, the
average fraction of hydrophobic amino acid residues in globular proteins is found to be close to the percolation threshold
of the effective lattice.
© 2002 John Wiley & Sons, Inc.
J Comput Chem 23: 166–171, 2002
Key words: percolation threshold; polymer chains; computer simulation; collapse transition; protein folding
Introduction
The behavior of dilute polymer systems under various solvent conditions has been the subject of an extensive research for many
years1 – 3 (see also discussions in refs. 4 and 5, and references
cited therein). Such systems have been commonly modeled by selfavoiding walks (SAWs) of N steps (bonds), i.e., N + 1 monomers
with self-attractions on a regular lattice; thus, two nonbonded
monamers that are nearest neighbors on the lattice gain energy
of interaction , ( = −||). With this model, a high absolute
temperature T corresponds to a good solvent conditions, where
the excluded volume interactions dominate the attractions and
the chain is swollen. Thus, the radius of gyration Rg scales as
Rg ∼ N ν , where ν = 3/4 = 0.75 in two dimensions6 (2d) and ν 0.59 in 3d (e.g., see ref. 7). However, as T is decreased (i.e., the
solvent conditions worsen) the attractions become more effective,
and at a temperature θ (the Flory θ temperature1 – 3 ) they cancel to
a large extent the excluded volume repulsions, and in 3d the chain
behaves in many respects like a Gaussian chain (e.g., νθ = 1/2);
in 2d, on the other hand,8 ν = 4/7 = 0.571. At T < θ the attractions prevail and the chain collapses, i.e., νc = 1/d. The degenerate
ground state is of maximal density and minimal surface to volume
ratio (see Figs. 1a and 1b). The collapse transition at θ has been
identified by de Gennes3 as a tricritical point with an upper critical
dimension three. It should be pointed out that at T slightly smaller
than θ only very long chains will collapse, whereas a short chain
will show a θ behavior, collapsing only at T , which is significantly
smaller that θ. The above values of ν have been approximately obtained also in the experiment.9 – 12
One may consider another SAW model in which only l, out
of the N + 1 monomers are attracting, i.e., their fraction is g =
l/(N +1); it is of interest to study how the value of g and the particular arrangement of the attracting monomers along the chain affect
the character of the transition and the ground state of the chain.
Indeed, such models have been employed to describe the behavior
of self-associating polymers in a good solvent, i.e., polymers that
attracting groups are attached to a small fraction of their monomers
(ionomers, for example). Joanny13 and Gates and Witten14 studied
analytically SAWs with a small fraction g of attracting monomers
in the θ and the strong excluded volume regimes, respectively. They
derived expressions that describe the polymer shape in terms of g
and other parameters and defined the conditions for which a col-
Correspondence to: H. Meirovitch; e-mail: [email protected]
Contract/grant sponsor: U.S. Department of Energy; contract/grant
number: DE-FG05-95ER62070
© 2002 John Wiley & Sons, Inc.
Polymer Collapse, Protein Folding, Percolation Threshold
Figure 1. Ground states of SAWs consisting of attracting H monomers
(full circle; their fraction is denoted by g) and neutral P monomers
(empty circles). Nonbonded nearest-neighbor H monomers interact
with negative energy = −|| (e.g., see arrows). (a) and (b): Different
ground states for g = 1 with energy 8. (c) The ground state for an
arrangement of a repeated group of succesive m = 2 H monomers
followed by 2n = 4 P monomers.
lapse occurs; however, they did not consider the arrangement of the
attraction monomers along the chain.
SAWs on a lattice with l attracting monomers have also been
studied extensively in the context of protein folding, starting with
the pioneering work of Gō and collaborators,15 which was followed
by others16 – 19 (for reviews on more recent work see refs. 20 – 24).
In particular, the present model is the HP model of Lau and
Dill,16, 25 where H denotes an attracting hydrophobic amino acid
residue and P is a polar residue, which is considered to be neutral;
for simplicity we shall use this notation as well.
In the present article we study the HP model on the square
and the simple cubic (SC) lattices using the scanning simulation
method.26, 27 We discuss the effect of g and the arrangement of
the H monomers along the chain on the ground state(s), the type
of transition, and the transition temperature. In particular, we are
interested in the case where the H monomers are distributed at
random. We provide theoretical arguments supported by simulation
results that a collapse transition will occur with high probability as
long as g > pc , where pc is the site percolation threshold for the
lattice studied. This conclusion is in accord with the fraction of
hydrophobic amino acids found in proteins.
Methods
Simulation of self-attracting SAWs at low temperatures with the
dynamical Metropolis method is inefficient because of the diffi-
167
culty to induce global conformational changes. In this respect the
scanning method is expected to perform better because the chains
are constructed step by step with the help of transition probabilities that depend on scanning f future steps. However, the chain
may get trapped in a dead end during construction; in this case it is
discarded, and the construction of a new SAW is started. Therefore,
the number of surviving chains is smaller than the number of chains
attempted. Also, the generated chains are not distributed according
to the Boltzmann probability; however, this bias can be removed
by importance sampling or equivalently by selecting an effectively
smaller set of accepted chains.27 The larger is f the higher the
efficiency, i.e., the larger is the number of the surviving chains as
well as the accepted ones. The scanning method has the advantage
that it provides the free energy as a by-product of the simulation.
In this study we apply f = 4 for SAWs on the square and the SC
lattices, which, however, requires scanning five steps ahead to take
into account also the attracting interactions. Because the excluded
volume interaction is much stronger and the attracting interaction
is much weaker for a SAW on the square than on the SC lattice
(maximum energy of 2 vs. 4 per H monomer, respectively), a
simulation with f = 4 is much more efficient on the SC than the
square lattice. Each simulation run is based on 3 × 105 attempted
SAWs, where results based on less than 104 accepted SAWs are not
considered.
The transition temperature for g < 1 is denoted by Tt to be
distinguished from θ defined for g = 1. We also use the notations
K = −/kB T and Kt for the reciprocal temperature and its transition value, respectively, where kB is the Boltzmann constant; the
shape exponent of the collapsed state (1/d) is denoted by νc . As
discussed later, for g < 1 the ground state might be rod-like or
layer-like in 2 and 3d, respectively, and even when a collapse occurs it is not clear whether a θ-point scaling behavior is guaranteed.
Also, even in the case of a θ-point transition, a highly accurate determination of Tt (for the infinite chain) from simulation data of
short chains would require large samples and a relatively complex
scaling analysis.28 Therefore, for the relatively short chains studied, Tt is defined as the temperature at which the radius of gyration,
Rg (N) scales as N 1/d ; this Kt is expected to overestimate significantly the corresponding Kθ temperature if it exists.
Results and Discussion
In Table 1 we present the analytical results1, 2, 8 for νθ and the most
accurate values of Kθ obtained for the square and the SC lattices
by computer simulation;4, 28, 29 as expected, Kθ decreases as d is
increased. These values of Kθ should be distinguished from those
for Kt in Table 2, obtained from the scaling of Rg as N 1/d . Thus,
for the SC lattice Kθ 0.27 in Table 1 is significantly smaller than
the value Kt = 0.45 obtained in Table 2 for g = 1.
Ordered Arrangements of the H Monomers
Let us first examine a SAW on a square lattice where m successive attracting monomers H are followed by an even number 2n
of nonattracting P monomers repeatedly (P1 P2 . . . P2n H1 H2 . . . Hm
P1 P2 . . . P2n . . .), i.e., g = m/(m + 2n). This system has a unique
ground state depicted in Figure 1c for m = 2 and n = 2, where
168
Meirovitch
•
Vol. 23, No. 1
•
Table 1. Critical Exponents for d = 2 and 3, Transition Temperatures,
and Site Percolation Thresholds for the Square and the SC Lattices.a
d
νθ
νc
Kθ = −/kB Tθ
pc
2
3
3
4/7b
1/2
1/2
1/2
1/3
1/3
0.658(4)c
0.274(6)d
0.2690(3)e
0.5927. . .
0.3116
0.3116
a The
percolation thresholds, pc are taken from ref. 30.
exact results of Duplantier and Saleur.8
c From ref. 28.
d From ref. 4.
e From ref. 29. The statistical error is given in parenthesis, e.g., 0.658(4) =
0.658 ± 0.004.
b The
every H monomer has the maximal number (2) of interactions, except for the 2m H monomers on the surface, which have only one
interaction. Thus, at low temperature a long enough chain will always become rod-like (with a width of m + 2n) for any value of m
and n, meaning that νc = 1 rather than 1/2, the value for a collapsed chain. However, a chain of length (m + 2n)2 has a collapsed
ground state. It should also be pointed out that a rod-like structure
will be obtained even for an extremely dilute concentration of the
H monomers (g → 0), where m = 1 and n is increased at will.
However, as n is increased (for constant number l of H monomers)
the entropy as well as the average energy increase (at a given temperature) meaning that the ground state becomes dominant only
at a very low temperature. Also, such a rod-like structure will be
difficult to generate by any computer simulation method because
its stabilization is based on large loops of size m + 2n; in particular, an efficient simulation with the scanning method would require
f ∼ m + 2n, which allows “sensing” at each step the existence of
the next H monomer. The other extreme case is the high concentration limit of H monomers, g → 1, where n = 2 and m is large.
Here, unlike the dilute limit, the energy dominates the entropy and
Table 2. The Reciprocal Transition Temperature Kt for Different Fractions
g of Randomly Selected H Monomers.a
d
νc
g
Kt = −/kB Tt
Nmax
2
2
2
3
3
3
3
3
3
1/2
1/2
1/2
1/3
1/3
1/3
1/3
1/3
1/3
0.8
0.7
0.6
1.0
0.8
0.6
0.5
0.4
0.32
1.2
1.4
1.5
0.45
0.6
0.9
1.4
2.0
2.0
90
70
70
210
180
160
100
100
90
a The dimension d = 2 and 3 refer to the square and the simple cubic
lattices, respectively. νc is the shape exponent for the collapsed chain, and
Nmax is the longest chain considered in the scaling analysis. For each g,
different arrangements based on different random number sequences lead
to different values of Kt ; therefore, the results for Kt are representative
values defined up to ±0.3.
Journal of Computational Chemistry
f = 5 or 6 would suffice for an efficient simulation with the scanning method. The ground state of these SAW models on a cubic
lattice is a two-dimensional layer of width m + 2n, i.e., νc = 1/2.
Simulations of SAWs of N ≤ 160 (m = 1 and n = 1, i.e.,
g = 1/3) on the square lattice have shown that at high T (low K)
the chain, as expected, is swollen, i.e., ν 0.75. However, when
K was increased, no sign of collapse was observed, i.e., ν was
not decreased but rather slightly increased. Thus, at the relatively
cold temperature, K = 3 we found ν 0.81, where at higher
K the scanning method (with f = 4) becomes inefficient; this
means that a rod-like structure was not obtained either. We, therefore, checked the simulated configurations by computer graphics
and found that they consisted basically of small “rods” (of the type
depicted in Fig. 1c) linked together by flexible segments, where
the length of these rods increases with increasing K. This suggests
that a transition occurs between a swollen and a rod-like structure but the transition is not sharp due to strong entropic effects,
which is expected for an one-dimensional model with short-range
interactions.31 We simulated this model (N ≤ 300) also on a
simple cubic lattice, where again a collapsed structure was not observed: As K was increased the corresponding values of ν were
decreased from ∼0.59 for small K to ∼0.5 for K = 1.9, 2, and 2.1
(at higher K the simulation becomes inefficient). Thus, these results
are in accord with a ground state that is a 2d layer, as discussed
earlier.
We also simulated on the square lattice the model m = 3 and
2n = 2 (g = 0.6) at various temperatures and again obtained that
even at the lowest temperature studied (K = 1.6), ν 0.78,
i.e., a rod-like structure was not obtained as yet. Indeed, from
free energy considerations the ground state is found to be unstable at K = 1.6 due to entropic effects: We obtain that the free
energy F for N = 100 is F /(NkB T ) = −1.207, which is significantly lower than Fgs /(NkB T ) = −0.944, the free energy of
the ground state alone at the same K (i.e., the entropy is zero and
Fgs /(NkT ) = KEgs /N, where Egs is the ground state energy).
In other words, the contribution of the entropy (∼0.757) to the free
energy at K = 1.6 is significant. The ground state of the cubic
lattice is a two-dimensional layer of width of five monomers. However, the structure of short chains (≤125) is expected to be compact.
Indeed, at K = 1 a collapse has been detected, that is, νc = 1/3
rather than 1/2. As in 2d, a transition to the ground state is also
expected here at a higher K.
One may argue that the structural properties discussed thus far
are not realistic because they reflect the geometrical restrictions
imposed by the square and the SC lattices (in which, e.g., two H
monomers separated by an odd number n of P monomers along the
chain cannot become nearest neighbors; thus, if n = 1, g = 0.5
with zero energy). However, the structure of Figure 1c (for m = 1,
n ≥ 1) is reminiscent of the α-helical rod-like structure occurring
frequently in polypeptides, which is mainly stabilized by shortrange hydrogen bonds. In fact, the helix-coil transition is not sharp,
and it can be described within the framework of the 1d Ising
model.31 Also, the loop created, for example, for n = 1 and m > 1
is similar to the a β-sheet structure prevalent in proteins, which
is based on short- and medium-range hydrogen bonds. Thus, even
though the ground states of our models on the cubic lattice are layers rather than rods or sheets, these models reflect the experimental
reality that the arrangement of the attracting monomers along the
Polymer Collapse, Protein Folding, Percolation Threshold
chain, their density g, and the type of interaction determine the
ground state that is not necessarily a collapsed structure.
Random Distribution of the H Monomers
In the examples studied thus far the attracting monomers were
arranged in a well-defined order along the chain; we now discuss
the case where they are distributed at random, i.e., a H (P) monomer
is determined with a probability g (1 − g) by a random number.
The probability of an arrangement with l monomers of type H is
g l (1 − g)(N +1−l) , and the probability to obtain l of them is defined
by the binomial distribution where the average is (N + 1)g and the
standard deviation σ = [(N + 1)g(1 − g)]1/2 . Thus, the typical
number of H monomers obtained in a random selection will pertain
to the range (N + 1)g ± σ and the chance that the corresponding arrangement is an ordered one is low. For large enough g the
ground state(s) is expected to be a collapsed (compact) structure,
and the question is whether a critical value gc exists, where for
g > gc a collapse is guaranteed (in the probabilistic sense discussed above). In what follows we argue that gc ≥ pc , where pc
is the site percolation threshold of the lattice studied. It should be
pointed out that in a typical percolation experiment each site of
a large empty lattice is visited and a monomer is placed there with
probability p. pc is the lowest probability at which percolation occurs, i.e., a cluster of monomers connecting the opposite sides of
the lattice is generated.30
Assume that a SAW on a square lattice is arranged in a perfectly compact configuration (e.g., the configurations depicted in
Figs. 1a and 1b); thus, if the H monomers are distributed at random,
the process becomes a percolation experiment, where for g ≥ pc
(pc 0.590 for the square lattice30 ) a percolating cluster defined
by the H monomers will be created. This cluster will connect the
opposite sides of the square structure, which most probably will
be “held” together by the attracting H monomers. The energy of
this cluster is low, because on average a nonsurface H monomer
will have close to 2dsquare − 2 nearest-neighbor nonbonded H
monomers, where dsquare = 1.896 is the fractal dimension of the
percolation cluster on the square lattice. Therefore, it is plausible
to assume that for g > pc randomly distributed H monomers will
lead to a collapsed degenerate ground state (notice that the energy
of the cluster will be different for different compact structures).
This argument can be extended to the simple cubic lattice where a
H monomer of the percolation cluster will have on average close
to 2dSC − 2 nearest-neighbor nonbonded H monomers; dSC = 2.5
is the fractal dimension of a percolation cluster on the SC lattice.
Because a collapsed structure is stabilized by short-, medium-, as
well as long-range interactions, the corresponding transition is expected to be sharp20 (notice that throughout this article the term
“sharp transition” means a transition that is significantly sharper
than a transition depending on short-range interactions, such as the
helix-coil transition discussed earlier).
Again, the probabilistic nature of this statement should be emphasized, meaning that specific distributions of H monomers with
g < pc can still lead to compact ground states, and as discussed
earlier, for g > pc , noncompact ground states are possible. Also,
compact structures that are not “perfect” (i.e., squares with the
maximal density in 2d, as those depicted in Figure 1a and b) can
also lead to νc = 1/2. Therefore, the percolation threshold should
169
be considered only as an approximate guiding value for collapse;
notice also that at g = pc the P monomers on the square lattice
will not percolate (1 − g = 0.4 < pc ), but they create a percolating
cluster on the SC lattice (1 − pc > pc ).
These theoretical considerations can be checked (even though
not proven) by computer simulation. We carried out a large number
of simulations as specified in the Methods section on the square and
SC lattices. For each value of g the arrangement of the H monomers
was determined at random, and simulations were performed at different temperatures to determine Kt at which νc 1/2 and 1/3 for
these lattices, respectively. Also, for each g, several arrangements
of the H monomers were tested based on different random number sequences. Table 2 demonstrates the expected increase in Kt
(i.e., the decrease in the temperature) as g is decreased (because
stronger “glue” between the interacting H monomers is needed to
hold the compact structure together, as their number is reduced and
entropic effects strengthen). It should be pointed out, however, that
each Kt (g) shown should be considered only as a representative
value obtained for a certain arrangement of the H monomers. We
have found that other random arrangements for a given g can lead to
Kt results that deviate by ±0.3 from the presented value; this stems
from both, the fluctuations in the number of monomers l discussed
earlier, and changes in the specific ordering of the H monomers
along the chain, an effect that increases with the decrease of g.
Such fluctuations are demonstrated in Figure 2, where results for
log(Rg2 /N) vs. log(N) obtained for g = 0.4 for the SC lattice are
plotted for different sequences of the H monomers together with
the corresponding best-fit lines. It is evident that the chain length
leading to efficient simulations (measured by the number of ac-
Figure 2. Log–log plots of the average radius of gyration, Rg2 /N
vs. the chain length N for three SAWs with different arrangements
of the H monomers on the SC lattice at the reciprocal temperature
Kt = −/KB T = 2.0. The H monomers of each chain were selected
at random with probability g = 0.4 but with different random number sequences. The best-fit lines denoted by full, dotted, and dashed
lines correspond to the circles, squares, and triangles, respectively; the
slopes of these lines are also provided where the collapsed value is
−νc = −1/3.
170
Meirovitch
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Vol. 23, No. 1
•
cepted chains), while relatively short, is sufficient to demonstrate
the existence of a collapse.
The most interesting result is that collapse transitions were detected very close to the percolation thresholds for g = 0.6 and
0.32 for the square and SC lattices, respectively, which supports the
main claim of this article. It should be emphasized, however, that
for the corresponding lattices and g values, out of four H arrangements only 3/4 and 2/4 arrangements have led to collapse; on the
other hand, collapse was obtained for specific arrangements based
on g = 0.5 and 0.24, respectively.
When g is much smaller than pc a collapsed ground state is
not expected. The most likely scenario is that several successive H
monomers along the chain that are separated by an even number
of monomers create local “blobs” separated (on the square lattice)
by flexible linear segments, and therefore, for large N, ν = 3/4.
Indeed, our simulations for g = 1/3 (random distribution) on the
square lattice led to ν ∼ 0.7 at both K = 2.5 and 2.8; however,
the corresponding values of G2 /N for N = 100, 0.77, and 0.70,
respectively, are significantly smaller than 1.09, obtained for SAW
without attractions at K = 0 (where ν is 0.75), which is because of
the effectively shorter self-attracting chain due to the blobs.
A continuum chain is expected to behave in many respect similar to the lattice models discussed above; however, for a continuum
chain the flexibility is not defined by the lattice but by geometrical restrictions induced by the intrachain interactions. Thus, a large
persistence length will prevent the formation of short- and mediumrange contacts between the attracting monomers but will have little
effect on the creation of long-range loops; in this respect the lattice is more restrictive, where H monomers separated by an odd
number of monomers along the chain can never become nearest
neighbors. When the attracting monomers appear in a specific order
along the chain, ordered structures can be formed, such as α-helices
and β-sheets, as has already been discussed. On the other hand, if g
is small and the attracting monomers are distributed at random, it is
difficult to predict the ground state(s), which, however, is expected
to become compact for a large enough g.
Because a continuum chain does not occupy a definite lattice, identification of g with a lattice percolation threshold is not
straightforward. However, for globular proteins a typical coordination number for an internal residue is32 7–8, i.e., between the
coordination numbers of the SC lattice (6) and the body-centered
cubic (BCC) lattice (8). Therefore, it is plausible to assume that the
percolation threshold for an effective protein lattice is between
the corresponding pc values, 0.3116 and 0.246, probably closer
to the latter. In refs. 33 and 34, based on various criteria, we have
classified as hydrophilic the nine residues, Asp, Glu, Gln, Asn, Lys,
Pro, Arg, Ser, and Thr, as neutral the amino acid His and as ambivalent, Ala, Gly, and Tyr. The seven remaining amino acids, Cys, Phe,
He, Leu, Met, Val, and Trp are the hydrophobic ones, which constitute 7/20 = 0.35 of the 20 occurring amino acids. However, their
fraction in the 19 smaller and larger proteins studied was found
to be 0.29, slightly above the effective percolation threshold and
thus in accord with our picture.
It should be pointed out, however, that the distribution of hydrophobic residues along a protein chain is not random but very
specific, as required for the stabilization of the native structure
(however, no clear correlation has been found between arrangements of hydrophobic residues in proteins from different families;
Journal of Computational Chemistry
in this respect the distribution of the hydrophobic residues can be
considered as random).
The hydrophobic residues avoid the contact with the surrounding water by concentrating in the interior of the protein structure.
If their fraction was much smaller than pc they would not be able
to percolate through the compact structure (assuming a random distribution), but they would cluster in smaller groups “wrapped” by
hydrophilic, neutral, or ambivalent residues. This, however, would
not be the most stable structure because the same degree of “coverage” (from water) of the hydrophobic residues also occurs in linear
structures consisting of blobs (see earlier discussion). In these
structures part of the interactions between the polar residues are
replaced by comparable polar–water interactions, while extra stability is gained from the increase in the chain entropy. Such local
coverage, however, is unlikely to occur for a percolating cluster
of hydrophobic residues, where the most stable structure is thus
a compact one. Therefore, a fraction pc of the hydrophobic residues
is necessary to guarantee compactness.
Analysis of protein structures has shown34 that the hydrophobic residues are not distributed homogeneously over the structure,
but their concentration in spherical layers around the center of
mass decreases significantly in going from the core towards the
surface of the protein, whereas an opposite trend is observed for
the hydrophilic residues. These changes in the distributions, however, stem partially from the fact that the structures of proteins
are not spherical but have ramified surfaces, and even internal
spherical layers might contain surface residues that preferably are
hydrophilic. To optimize the electrostatic energy the hydrophilic
residues in each layer are arranged in localized clusters, where the
fraction of these residues is larger than that in the entire layer; this
induces similar clustering of the hydrophobic residues as well.35
Therefore, it is very likely that both types of residues percolate
through the protein structure. In this context it should be pointed out
that while the clustering of the H (P) monomers in the simplified
HP model is induced by the HH attractions, the model still reflects
the global features of the structural organization of proteins, and
the present study, therefore, sheds new light on the hydrophobicitydriven mechanism of protein collapse in terms of the percolation
theory.
Summary
We have studied the structural transition of polymers with attracting monomers in the dilute regime from a swollen shape at
high temperatures to the ground state (which might be degenerate) at low temperatures. Extensive simulations of the HP model
on the square and SC lattices were carried out with the scanning method, where the criterion for a structural change is the
change in the shape exponent ν, describing the scaling of the radius of gyration with the chain length. As expected, the ground
state depends on the lattice, the fraction of the H monomers, and
their specific arrangement along the chain, which can lead to rodlike (ν = 1) and 2d layer-like (ν = 1/2) ground states on
the square and the SC cubic lattices, respectively. Because these
ground states are “held” by short- and medium-range interactions
the corresponding transitions are not expected to be sharp but
to occur through a range of temperatures. Whereas these ground
Polymer Collapse, Protein Folding, Percolation Threshold
states are lattice dependent, they reflect the experimental reality
that ordered structures, such as α helices and β sheets appear frequently in proteins and polypeptides due to orderly placed donor
and acceptor groups of hydrogen bonds along the chain. When
the H monomers are distributed at random and their fraction g is
larger than the site percolation threshold of the lattice, a collapsed
ground state (which can be degenerate) with a sharp transition
is expected. This conclusion, drawn for lattice models, also applies to globular proteins where the residues can approximately
be described as occupying an effective lattice with coordination
number between the coordination numbers of the SC and the BCC
lattices; hence, with an intermediate percolation threshold. This
threshold, indeed, is very close the average fraction of hydrophobic
residues in proteins. Thus, the percolation theory applied to the HP
model sheds new light on the hydrophobicity-driven protein collapse.
Acknowledgments
I thank Professor I. Rabin for valuable discussions, and Dr. B. Das
for her help.
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